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A192252 0-sequence of reduction of (n!) by x^2 -> x+1.

%I A192252 #16 Jul 14 2022 17:25:09
%S A192252 1,1,3,9,57,417,4017,44337,568497,8188977,131568177,2326992177,
%T A192252 44958134577,941649129777,21254190979377,514247427715377,
%U A192252 13277149259395377,364340640790147377,10588931448837763377,324919870905259651377,10496883167091791491377
%N A192252 0-sequence of reduction of (n!) by x^2 -> x+1.
%C A192252 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%C A192252 After the tenth term, the final digit is 7, for terms in both A192252 and A192253. After the 100th term, the final 6 digits of each term of A192252 are 9,3,1,3,7,7.
%F A192252 Conjecture: a(n) +(-n-1)*a(n-1) -n*(n-2)*a(n-2) +n*(n-1)*a(n-3)=0. - _R. J. Mathar_, May 04 2014
%F A192252 Conjecture: a(n) = Sum_{k=0..n} A052554(k). - _Sean A. Irvine_, Jul 14 2022
%e A192252 The sequence (n!)=(1,1,2,6,24,120,...) provides coefficients for the power series 1+x+2x^2+6x^3+..., of which the (n+1)st partial sum is the polynomial p(x)=1+x+2x^2+...+(n!)x^n, of which reduction by x^2 -> x+1 (as presented at A192232) is A192252(n)+x*A192253(n).
%t A192252 c[n_] := n!; (* A000142 *)
%t A192252 Table[c[n], {n, 1, 15}]
%t A192252 q[x_] := x + 1;
%t A192252 p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n]
%t A192252 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192252 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 50}]
%t A192252 Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}]  (* A192252 *)
%t A192252 Table[Coefficient[Part[t, n], x, 1], {n, 1, 50}]  (* A192253 *)
%t A192252 Table[Coefficient[(-7 + Part[t, n])/10, x, 0], {n, 1, 30}]
%t A192252 (* _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192252 Cf. A192232, A192253.
%K A192252 nonn
%O A192252 0,3
%A A192252 _Clark Kimberling_, Jun 27 2011